Let us consider the real numbers $x , y , t$ and $u$ satisfying the following four conditions:
$$\begin{aligned} & y \geqq | x | \\ & x + y = t \\ & x ^ { 2 } + y ^ { 2 } = 12 \\ & x ^ { 3 } + y ^ { 3 } = u \end{aligned}$$
We are to find the ranges of values which $t$ and $u$ can take.
(1) From (1) and (3), we see that the point $( x , y )$ is located on the arc which is a quadrant of the circle having its center at the origin and the radius $\mathbf { A }$. Moreover, the coordinates of the end points of this arc are
$$( \sqrt { \mathbf { C } } , \sqrt { \mathbf { CD } } ) \text { and } ( - \sqrt { \mathbf { C } } , \sqrt { \mathbf { D } } ) .$$
From this and (2), we also see that the range of values which $t$ can take is
$$\mathbf { E } \leqq t \leqq \mathbf { F } . \mathbf { G } .$$
(2) Next, from (2) and (3), we have
$$x y = \frac { \mathbf { H } } { \mathbf { I } } \left( t ^ { 2 } - \mathbf { J K } \right)$$
and further, using (4) we also have
$$u = \frac { \mathbf { L } } { \mathbf{L}} \left( \mathbf{NO} \, t - t ^ { 3 } \right)$$
Hence, since
$$\frac { d u } { d t } = \frac { \mathbf { P } } { \mathbf { Q } } \left( \mathbf { RS } - t ^ { 2 } \right)$$
the range of values which $u$ can take under the condition (5) is
$$\mathbf { TL } \leqq u \leqq \mathbf { UV } \sqrt { \mathbf{ W } } .$$